Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Related questions
Question
![**Title: Permutations and Divisibility**
**Objective:**
Prove that if \( r \leq s \leq n \), then \( P(n, s) \) is divisible by \( P(n, r) \).
----
**Explanation:**
The problem involves proving a property related to permutations. Here, \( P(n, s) \) represents the number of permutations of \( n \) items taken \( s \) at a time. Similarly, \( P(n, r) \) represents permutations of \( n \) items taken \( r \) at a time.
**Steps to Approach the Proof:**
1. **Understand Permutations:**
- The formula for permutations \( P(n, k) \) is given by:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
- This formula represents selecting \( k \) items from \( n \) without replacement and considering the order.
2. **Divisibility Condition:**
- To prove that \( P(n, s) \) is divisible by \( P(n, r) \), show that:
\[
\frac{P(n, s)}{P(n, r)}
\]
is an integer.
3. **Simplifying the Expressions:**
- Write the expressions for \( P(n, s) \) and \( P(n, r) \):
\[
P(n, s) = \frac{n!}{(n-s)!}
\]
\[
P(n, r) = \frac{n!}{(n-r)!}
\]
- Compute the ratio:
\[
\frac{P(n, s)}{P(n, r)} = \frac{(n-r)!}{(n-s)!}
\]
4. **Prove the Ratio is Integer:**
- Note that since \( r \leq s \), each step from \( (n-r)! \) to \( (n-s)! \) is a product of consecutive integers:
\[
(n-s+1)(n-s+2)\cdots(n-r)
\]
- The product is clearly an integer, confirming divisibility.
This completes the proof that \( P(n, s) \) is divisible by \( P(n, r) \) under the given conditions.](https://content.bartleby.com/qna-images/question/b48054c1-c883-40e2-8364-5c805e917eb2/66b02d4b-a13f-424b-978e-2f2a8c84a7af/qvub9al_processed.jpeg)
Transcribed Image Text:**Title: Permutations and Divisibility**
**Objective:**
Prove that if \( r \leq s \leq n \), then \( P(n, s) \) is divisible by \( P(n, r) \).
----
**Explanation:**
The problem involves proving a property related to permutations. Here, \( P(n, s) \) represents the number of permutations of \( n \) items taken \( s \) at a time. Similarly, \( P(n, r) \) represents permutations of \( n \) items taken \( r \) at a time.
**Steps to Approach the Proof:**
1. **Understand Permutations:**
- The formula for permutations \( P(n, k) \) is given by:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
- This formula represents selecting \( k \) items from \( n \) without replacement and considering the order.
2. **Divisibility Condition:**
- To prove that \( P(n, s) \) is divisible by \( P(n, r) \), show that:
\[
\frac{P(n, s)}{P(n, r)}
\]
is an integer.
3. **Simplifying the Expressions:**
- Write the expressions for \( P(n, s) \) and \( P(n, r) \):
\[
P(n, s) = \frac{n!}{(n-s)!}
\]
\[
P(n, r) = \frac{n!}{(n-r)!}
\]
- Compute the ratio:
\[
\frac{P(n, s)}{P(n, r)} = \frac{(n-r)!}{(n-s)!}
\]
4. **Prove the Ratio is Integer:**
- Note that since \( r \leq s \), each step from \( (n-r)! \) to \( (n-s)! \) is a product of consecutive integers:
\[
(n-s+1)(n-s+2)\cdots(n-r)
\]
- The product is clearly an integer, confirming divisibility.
This completes the proof that \( P(n, s) \) is divisible by \( P(n, r) \) under the given conditions.
Expert Solution
arrow_forward
Step 1
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Similar questions
- How do I do this? Detailed help will be appreciated!arrow_forwardProb q6arrow_forwardLet x≥2, conjecture and prove an expression with no summation signs and no recursviely calculated values for S(x, 2), and S(x, x −2). Suppose x≥1, conjecture and prove an expression with no summation signs and no recursviely calculated values for c(x, x-1) note: S(x, x-1) = (x choose 2) this is a combinatorics math problemarrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Algebra and Trigonometry (6th Edition)AlgebraISBN:9780134463216Author:Robert F. BlitzerPublisher:PEARSON
Contemporary Abstract AlgebraAlgebraISBN:9781305657960Author:Joseph GallianPublisher:Cengage Learning
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning 
Algebra And Trigonometry (11th Edition)AlgebraISBN:9780135163078Author:Michael SullivanPublisher:PEARSON
Introduction to Linear Algebra, Fifth EditionAlgebraISBN:9780980232776Author:Gilbert StrangPublisher:Wellesley-Cambridge Press
College Algebra (Collegiate Math)AlgebraISBN:9780077836344Author:Julie Miller, Donna GerkenPublisher:McGraw-Hill Education
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:9781305657960
Author:Joseph Gallian
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:9780135163078
Author:Michael Sullivan
Publisher:PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:9780980232776
Author:Gilbert Strang
Publisher:Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:9780077836344
Author:Julie Miller, Donna Gerken
Publisher:McGraw-Hill Education